A190965 a(n) = 4*a(n-1) - 6*a(n-2), with a(0)=0, a(1)=1.

0, 1, 4, 10, 16, 4, -80, -344, -896, -1520, -704, 6304, 29440, 79936, 143104, 92800, -487424, -2506496, -7101440, -13366784, -10858496, 36766720, 212217856, 628271104, 1239777280, 1189482496, -2680733696, -17859829760, -55354916864, -114260688896

Offset

0,3

Comments

For the quaternion Q = 2+j+k, Q^n = r(n) + a(n)*(j+k). The sequence of real-parts r(n) is A266046. - Stanislav Sykora, Dec 20 2015

Links

Stanislav Sykora, Table of n, a(n) for n = 0..1000

Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.

Index entries for linear recurrences with constant coefficients, signature (4,-6).

Formula

G.f.: x/(1-4*x+6*x^2). - Philippe Deléham, Oct 12 2011

2*a(n)^2 + A266046(n)^2 = 6^n. - Stanislav Sykora, Dec 20 2015

From G. C. Greubel, Jan 10 2024: (Start)

a(n) = 6^((n-1)/2)*ChebyshevU(n-1, sqrt(2/3)).

E.g.f.: (1/sqrt(2))*exp(2*x)*sin(sqrt(2)*x). (End)

Mathematica

LinearRecurrence[{4, -6}, {0, 1}, 50]

Prog

(PARI) a(n)=([0, 1; 0, 0]*[0, -6; 1, 4]^n)[1, 1] \\ Charles R Greathouse IV, May 31 2011
(Magma) [n le 2 select n-1 else 4*Self(n-1) -6*Self(n-2): n in [1..41]]; // G. C. Greubel, Jan 10 2024
(SageMath)
A190965=BinaryRecurrenceSequence(4, -6, 0, 1)
[A190965(n) for n in range(41)] # G. C. Greubel, Jan 10 2024

Crossrefs

Cf. A190958 (index to generalized Fibonacci sequences).

Cf. A088137 (Inv. Bin. Trans.), A168175, A213421, A266046.

Sequence in context: A259262 A218211 A302197 * A099457 A055103 A285629

Adjacent sequences: A190962 A190963 A190964 * A190966 A190967 A190968

Keyword

sign,easy

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Status

approved